Formation of nonlinear modes in one-dimensional quasiperiodic lattices with a mobility edge

Dmitry A. Zezyulin; Georgy L. Alfimov

REVIEW 3 major objections 5 minor 87 references

Formation of nonlinear modes in one-dimensional quasiperiodic lattices with a mobility edge

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read In a one-dimensional Bose-Einstein condensate with a bichromatic quasiperiodic potential, nonlinear localized modes are born through pitchfork or saddle-node bifurcations dictated by the band-edge structure, with the mobility edge…

desk verdict Useful numerical bifurcation study that needs a convergence check before the above-mobility-edge claims (pitchfork, square-root N(mu)) are trusted. read the letter →

arxiv 2411.13936 v1 pith:7XUA3O5I submitted 2024-11-21 nlin.PS cond-mat.quant-gasphysics.optics

Dmitry A. Zezyulin , Georgy L. Alfimov This is my paper · ORCID

classification nlin.PScond-mat.quant-gasphysics.optics MSC 35Q5537G1037G15

keywords Bose-Einsteincondensatequasiperiodiclatticemobilityedgenonlinearmodespitchforkbifurcationsaddle-nodeGross-Pitaevskiiequationapproximantpath

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper studies how steady nonlinear modes are born from the linear spectrum in a one-dimensional Bose-Einstein condensate with repulsive interactions trapped in a bichromatic quasiperiodic potential. Working with rational approximants of the golden-ratio potential, it finds that the mobility edge splits the formation mechanism in two: below it, the linear mode is already localized and the atom number grows linearly with chemical potential, $N(\mu)\approx (\mu-\tilde{\mu}_n)/\tilde{\chi}_n$; above it, an extended linear mode gradually localizes and $N(\mu)$ follows a square-root-like law. In a symmetric potential, nonlinear modes are born through both pitchfork and saddle-node bifurcations that mimic symmetric and asymmetric double wells, while a nonzero phase shift between the lattices produces a cascade of saddle-node bifurcations. If these patterns are robust, they connect the fractal linear spectrum and the mobility edge directly to the multiplicity of nonlinear localized states and give experimental predictions for atom-number scaling.

What carries the argument

The argument is carried by the approximant path: replace the irrational frequency ratio $\phi$ by a Fibonacci fraction $p/q$, restrict the problem to the periodic cell $I_q=[-\pi q/2,\pi q/2)$ with periodic boundary conditions, and solve the Gross-Pitaevskii equation $\mu\psi=H\psi+g\psi^3$ by Fourier collocation and Newton's method. The organizing device is a classification of right band edges as symmetric-antisymmetric pairs, boundary-localized modes, or isolated symmetric modes. The bifurcation analysis rests on the perturbation formula $\psi(x)\approx((\mu-\tilde{\mu}_n)/\tilde{\chi}_n)^{1/2}\tilde{\psi}_n(x)$, which yields $N(\mu)\approx(\mu-\tilde{\mu}_n)/\tilde{\chi}_n$, where $\tilde{\chi}_n$ is the inverse participation ratio; this formula is what makes below-the-edge branches linear and above-the-edge branches steep and root-like.

What would settle it

Repeat the bifurcation analysis at the next Fibonacci approximants ($p/q=144/89$, $233/144$, $377/233$) or with a different irrational frequency ratio and check whether the predicted patterns persist: a symmetric-antisymmetric edge that changes to a boundary edge and turns a pitchfork into a different bifurcation, or an above-edge $N(\mu)$ branch that deviates from the square-root law, would refute the claimed universality.

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Extended reading notes

Core claim

For the symmetric bichromatic potential $V(x)=v_1\cos(2x)+v_2\cos(2\phi x)$ with $\phi$ the golden ratio, the right edge of a linear band can be a symmetric-antisymmetric pair, a single strongly localized symmetric mode, or a mode pinned to the domain boundary. The paper shows that each edge structure dictates a distinct nonlinear scenario: symmetric-antisymmetric pairs trigger pitchfork symmetry-breaking bifurcations whose particle number is set by the ratio of the eigenvalue splitting to the inverse participation ratio; isolated modes hybridize with nearby states to form in-phase and out-of-phase bound states through saddle-node bifurcations; and above the mobility edge, extended linear states develop into gap-soliton-like families with a root-like $N(\mu)$ curve. With a nonzero inter-lattice phase shift, the potential is generically asymmetric, the special eigenvalue pairs disappear, and every linear mode produces a nonlinear family that meets the next eigenvalue, yielding a chain of saddle-node bifurcations before localized modes reach the gap. The central claim is that the mobility edge and the Fibonacci-labelled gap structure of the linear spectrum organize which bifurcation pattern applies.

Load-bearing premise

The conclusions are drawn from finite rational approximants of the golden-ratio potential with periodic boundary conditions, and the paper assumes this approximant path faithfully represents the true infinite quasiperiodic system even though Table I shows the band-edge structure switching between approximants.

Editorial extensions

If this is right

Below the mobility edge, atom number grows linearly with chemical potential, $N(\mu)\approx (\mu-\tilde{\mu}_n)/\tilde{\chi}_n$, so measuring $N(\mu)$ near a band edge gives the inverse participation ratio of the underlying linear mode.
Above the mobility edge, an initially extended linear mode gradually localizes as $\mu$ moves into the gap, producing a square-root-like $N(\mu)$ branch characteristic of gap solitons in periodic media.
In a symmetric bichromatic potential, band edges formed by symmetric-antisymmetric pairs produce pitchfork symmetry breaking at a particle number set by the eigenvalue splitting divided by the inverse participation ratio; isolated or hybridizing edge modes produce saddle-node bifurcations and in-phase/out-of-phase bound states.
A nonzero phase shift between the two lattices removes the symmetry, converting pitchforks into a cascade of saddle-node bifurcations in which each linear mode contributes a localized branch that reaches the gap.
Because the Gross-Pitaevskii equation is mathematically equivalent to the nonlinear Schrödinger equation for paraxial light, the same bifurcation patterns should appear as optical mode formation in quasiperiodic photonic lattices.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper's finite-approximant numerics, the setup suggests that in the true quasiperiodic limit the asymmetric case may exhibit an infinite accumulation of saddle-node bifurcations near a band edge, a property the paper does not directly establish.
The Fibonacci-labelled gaps are special to the golden ratio; for other irrational ratios the universal dichotomy (pitchfork for symmetric, saddle-node cascade for asymmetric) may persist even though the ordering of modes at band edges changes, so testing another ratio would separate the general mechanism from golden-ratio-specific structure.
In the optical realization, the square-root $N(\mu)$ law above the mobility edge translates into a measurable power-versus-propagation-constant curve for soliton formation in quasiperiodic photonic lattices, and the predicted symmetry-breaking particle number $N_{\mathrm{SB}}\approx(\tilde{\mu}_{55}-\tilde{\mu}_{54})/\tilde{\chi}_{55}$ could be checked directly in a cold-atom experiment.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Bottom line: a clear, competent numerical bifurcation study that will be useful, but the claims about the above-mobility-edge regime rest on one rational approximant and need a convergence check before I treat them as established.

What's actually new: the paper extends the authors' prior program (Ref. [50], symmetry breaking at the left band edge) to right band edges and to modes above the mobility edge, in a symmetric bichromatic potential with v1=v2=0.8. It classifies right-edge structures in terms of symmetric-antisymmetric pairs, boundary-localized modes, and a single robust symmetric mode, and shows numerically that below the mobility edge N(mu) is linear while above it looks square-root-like. It also shows, for one nonzero phase shift, a cascade of saddle-node bifurcations. The methods are standard (Fourier collocation with 4500 harmonics, Newton, linear stability), the perturbation formulas (10)-(11) and the two-mode estimate (12) are checked against the numerics, and the paper is honest about the gap-E caveats. That is genuinely useful, especially for experimental groups planning quasiperiodic BEC or photonic experiments.

Soft spots. The approximant path is the load-bearing weakness. Table I shows the right-edge structure of gaps A, B, D changing type between 's-a/s pair' and 'boundary' as q increases; gap D, the flagship below-edge pitchfork example, becomes a boundary edge at p/q=144/89. Above the mobility edge, the gap E 'pair of virtually coinciding extended eigenfunctions' has spacings ~1e-8 and ~1e-11 at 55/34 and 89/55, comparable to numerical precision, and at 144/89 the right edge is a single extended eigenfunction. All nonlinear diagrams in gap E are computed only at 89/55. So the most novel claim - that the mobility edge dichotomizes the bifurcation structure and the scaling of N(mu) - is not yet robust in the infinite-system limit; it may be sensitive to which approximant is selected. The generic asymmetric case is one example at one theta. No code, data, or explicit tolerances are provided, which makes independent verification harder.

Does the central argument hold? The below-edge picture - linear N(mu) and pitchfork/saddle-node from localized double-well pairs - is well supported. The above-edge picture is plausible but not yet established. The paper overclaims slightly with 'general patterns' given the limited approximant data.

Who for: researchers in quasiperiodic lattices, nonlinear waves, BECs. A serious referee could add value by asking for a convergence study and a second approximant for the nonlinear runs. I'd accept it for peer review, and I'd cite it for the classification and below-edge results with a caveat.

Referee Report

3 major / 5 minor

Summary. The paper studies nonlinear steady states of the one-dimensional Gross-Pitaevskii equation with a bichromatic quasiperiodic potential, using rational approximants of the golden-ratio frequency ratio with periodic boundary conditions. The authors classify the right band edges of selected spectral gaps (A–E) and compute bifurcation diagrams of nonlinear modes near those gaps for the approximant p/q = 89/55 with lattice amplitudes v1 = v2 = 0.8. They report symmetry-breaking pitchfork bifurcations for near-degenerate symmetric/antisymmetric pairs below the mobility edge, saddle-node bifurcations due to hybridization of modes with different spatial profiles, a different (square-root-like) N(mu) behavior above the mobility edge, and a cascade of saddle-node bifurcations for a nonzero phase shift. The central claim is that the mobility edge dichotomizes the formation of nonlinear modes: linear N(mu) dependence below, square-root-like above, with the bifurcation taxonomy mimicking symmetric and asymmetric double-well potentials.

Significance. If established, the claimed connection between the fractal linear spectrum, the mobility edge, and the bifurcation structure of nonlinear localized modes would be a useful organizing principle for quasiperiodic BECs and photonic lattices, with clear experimental relevance. The numerical methodology is standard and adequately described: Fourier collocation with 4500 harmonics, Newton's method, and linear-stability analysis via Eq. (8). The perturbation formulas (10)–(11) are standard and Eq. (12) is explicitly checked against numerics. The numerical results are obtained by direct solution of the nonlinear PDE, so there is no circularity in the bifurcation detection. The main weakness is the reliance on a single rational approximant for all nonlinear bifurcation diagrams, while the band-edge classification itself visibly changes across approximants in Table I. This makes the central claims about universal patterns and the role of the mobility edge not yet fully established in the infinite-quasiperiodic limit.

major comments (3)

[Sec. III, Table I] The manuscript's central claim of universal patterns is undermined by the approximant dependence of the band-edge classification. Table I shows that the same gap changes its right-edge structure as p/q increases: gap A switches between 's-a/s pair', 'boundary', and 'new s-a/s pair'; gap B switches between 'boundary' and 's-a/s pair'; and gap D, the flagship below-mobility-edge pitchfork example in Fig. 4, becomes a 'boundary' edge at p/q = 144/89. Since all nonlinear bifurcation diagrams in Sec. IV are computed only for p/q = 89/55, the paper does not demonstrate that the pitchfork-vs-saddle-node taxonomy is the infinite-system behavior rather than an artifact of that particular approximant. The authors should either compute the relevant bifurcation diagrams for at least one more approximant (e.g., 55/34, 144/89, or 233/144) for gaps C, D, and E, or provide a convergence argument showing that the bifurcation type stabilizes for sufficiently large q.
[Sec. IV D, Fig. 6] The above-mobility-edge analysis for gap E rests on numerically fragile data. The band edge at p/q = 89/55 is formed by a pair of eigenvalues with splitting ~10^-11, which the authors themselves note is comparable to numerical error, and for p/q = 144/89 the edge is instead a single extended eigenfunction well separated from the coincident pair. The symmetry-breaking bifurcation and the square-root-like N(mu) law shown in Fig. 6 are computed only at 89/55. This does not establish the claimed dichotomy between below- and above-mobility-edge bifurcation scenarios in the quasiperiodic limit. A direct convergence test (e.g., increasing the number of Fourier harmonics for the near-degenerate pair, or repeating the gap-E bifurcation analysis at another approximant) is needed before the central claim can be accepted.
[Sec. IV A and Sec. IV D] The asserted square-root law for N(mu) above the mobility edge is not actually derived or quantitatively verified. Equation (11) gives a linear dependence N ≈ (mu - mu_n)/chi_n for a bifurcation from an isolated linear mode with fixed IPR. For the extended modes above the mobility edge, chi is not constant along the family, so a different law is expected, but the paper only states that the dependence 'resembles the root-law behavior' and attributes it to similarity with gap solitons in periodic media (Sec. IV D). To make the 'linear below, square-root above' dichotomy load-bearing, the authors should provide a quantitative check (e.g., a log-log fit of N(mu) over the gap, or a reduced-model derivation) and show that it holds for at least one additional approximant.

minor comments (5)

[Sec. II, Eq. (8)] Typographical errors: 'eigenvalue' appears as 'and eigenvalue' and 'the the remaining' appears in the enumeration of spectral properties. These should be corrected.
[Sec. IV A] The phrase 'nonlinear modes theta bifurcate near those bands' in the opening sentence of Sec. IV A should read 'that bifurcate'.
[Fig. 5] The caption of Fig. 5 is dense and the labels SB1, SB2, SN1, SN2 are not all explicitly explained in the caption; a short list of which line corresponds to which family would improve readability.
[Sec. III, Table I] The caption of Table I states 'the structure of the right band edges adjacent to gaps A–D from Fig. 1' but does not mention that gap E is discussed in the text; adding this would help the reader locate the full classification.
[Sec. IV E] The asymmetric case is illustrated for a single arbitrarily chosen phase shift theta ≈ 1.7π in gap B. While the cascade of saddle-node bifurcations is plausible, a brief statement about whether θ = 1.7π is generic (e.g., not accidentally restoring symmetry) would strengthen the presentation.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the bifurcation results are obtained by direct numerical solution of the Gross-Pitaevskii equation, with perturbation formulas used only as ancillary checks.

full rationale

The paper's central claims are produced by direct simulation of the stationary Gross-Pitaevskii equation (Eq. 3) using Newton's method with periodic boundary conditions, not by fitting or assuming the target bifurcation taxonomy. The perturbation formulas (10)-(11) are standard asymptotic expansions used to interpret the slopes of the N(mu) curves, and the two-mode estimate (12) is explicitly compared with the full numerics: the paper reports 'the approximation (12) gives NSB ~ 6e-4, which is in reasonable agreement with the numerical value NSB ~ 3e-4,' and then discusses the discrepancy. Thus that estimate is a consistency check, not the source of the prediction. The only self-citations, Refs. [50] and [63], supply previous two-mode Josephson-junction and perturbation-theory results that are independently checkable from the equations and are not load-bearing for the new bifurcation classifications. The finite-approximant route and periodic boundary conditions are an approximation strategy stated in Sec. II; Table I does show variation of band-edge structure across rational approximants, but that is a robustness or convergence concern about the infinite-quasiperiodic-system interpretation, not a circular step. No equation is defined in terms of the claimed conclusion, no fitted parameter is relabeled as a prediction, and no uniqueness theorem is imported to force the model choice. Accordingly, no self-definitional, fitted-input, ansatz-smuggling, self-citation-chain, or renaming circularity is present.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

No new physical entities are introduced. The central claims rest on three main supports: standard numerical methods for the GPE, the approximant-path assumption that finite periodic approximants capture the infinite quasiperiodic system, and the double-well analogy used to interpret the bifurcation patterns. The parameters v1=v2=0.8, p/q=89/55, and theta about 1.7 pi are fixed input choices rather than fitted constants, but they bear a large part of the demonstrative weight.

free parameters (4)

Lattice amplitudes v1, v2 = v1 = v2 = 0.8
All nonlinear computations use this fixed amplitude combination; the paper does not scan v1, v2 to show that the claimed patterns persist across parameter space.
Rational approximant p/q = 89/55 for the main nonlinear results
The nonlinear bifurcation diagrams in Sec. IV use p/q = 89/55, while Table I shows that some band-edge structures change between approximants, so the universality of the patterns depends on this choice.
Phase shift theta = theta about 1.7 pi for the asymmetric case
The 'generic' cascade of saddle-node bifurcations is demonstrated at a single phase shift chosen arbitrarily, not over a range of theta.
Frequency ratio phi = golden ratio (1+sqrt(5))/2
The golden ratio is chosen to exploit Fibonacci rational approximants; it is a model choice rather than a fit, but the conclusions are established only for this ratio.

assumptions (5)

domain assumption The approximant path with periodic boundary conditions on a finite domain of length pi q faithfully represents the infinite quasiperiodic system for the bifurcation structure studied here.
Invoked when V(x) is replaced by V_q(x) and boundary conditions (7) are imposed in Sec. II; the robustness claims in Sec. III rely on this.
standard math The perturbation expansion of Eq. (10), psi(x) about sqrt((mu - mu_tilde_n)/chi_tilde_n) psi_tilde_n(x), is valid near band edges.
Used to derive the N(mu) scaling in Eq. (11), which underpins the distinction between below- and above-mobility-edge behavior; taken from Ref. [63].
domain assumption The two-mode reduction giving Eq. (12), NSB about (mu_tilde_55 - mu_tilde_54)/chi_tilde_55, is accurate enough to estimate the symmetry-breaking threshold.
Used to compare with the numerically found NSB; the authors note a factor-of-two discrepancy, showing the assumption is approximate.
domain assumption The Gross-Pitaevskii equation is a valid mean-field description of the trapped Bose-Einstein condensate.
The model in Eq. (1) is the starting point of the entire study; validity of mean-field theory is assumed.
ad hoc to paper The symmetric bichromatic potential of Eq. (2) is sufficiently analogous to a symmetric double well to license double-well bifurcation phenomenology.
The analogy is used explicitly in Secs. II and IV to interpret pitchfork and saddle-node bifurcations; it relies on the specific potential shape with a maximum at x = 0.

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Cite this review

Pith. "Pith review of Formation of nonlinear modes in one-dimensional quasiperiodic lattices with a mobility edge." pith.science (2026). https://pith.science/paper/7XUA3O5I

@misc{pith2026241113936,
  author       = {Pith},
  title        = {Pith review of: Formation of nonlinear modes in one-dimensional quasiperiodic lattices with a mobility edge},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7XUA3O5I}},
  note         = {Machine review of arXiv:2411.13936}
}

read the original abstract

We investigate the formation of steady states in one-dimensional Bose-Einstein condensates of repulsively interacting ultracold atoms loaded into a quasiperiodic potential created by two incommensurate periodic lattices. We study the transformations between linear and nonlinear modes and describe the general patterns that govern the birth of nonlinear modes emerging in spectral gaps near band edges. We show that nonlinear modes in a symmetric potential undergo both symmetry-breaking pitchfork bifurcations and saddle-node bifurcations, mimicking the prototypical behaviors of symmetric and asymmetric double-well potentials. The properties of the nonlinear modes differ for bifurcations occurring below and above the mobility edge. In the generic case, when the quasiperiodic potential consists of two incommensurate lattices with a nonzero phase shift between them, the formation of localized modes in the spectral gaps occurs through a cascade of saddle-node bifurcations. Because of the analogy between the Gross-Pitaevskii equation and the nonlinear Schr\"odinger equation, our results can also be applied to optical modes guided by quasiperiodic photonic lattices.

Figures

Figures reproduced from arXiv: 2411.13936 by the authors.

**Figure 1.** (a) Chemical potentials ˜µ1 < µ˜2 < . . . µ˜n < . . . and (b) IPRs ˜χn for the 100 lowest linear modes, computed from the linear eigenvalue problem (4) with a rational approximation p/q = 89/55, and amplitudes of the lattices v1 = v2 = 0.8. Several gaps are labelled with letters A-E in the upper panel. The gaps are located at mode numbers n corresponding to the Fibonacci sequence, as highlighted by dotted vertical … view at source ↗

**Figure 2.** Schematic illustrations for structure of right ba [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Eigenfunctions at the right edges of bands adja [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: The number of particles N and IPR χ are plotted against the chemical potential µ for the nonlinear modes that emerge in gap D from linear modes with numbers n = 54 and n = 55. Panels (a,b) show a closer view of the band edge where the nonlinear families are born, while…

**Figure 6.** Figure 6: The number of particles N (a,c) and the IPR χ (b,d) vs. the chemical potential µ for the nonlinear modes emerging in gap E. Panels (a,b) present a closer-up view of the band edge where the nonlinear families are formed, while panels (c,d) provide a smaller-scale pictur…

**Figure 5.** Figure 5: The number of particles N (a) and the IPR χ (b) vs. µ for the nonlinear families in gap C. The yellow circles in (a) indicate the chemical potentials ˜µn of the noninteracting BEC. Labels ‘SB1,2’ and ‘SN1,2’ indicate two symmetrybreaking and two saddle-node bifurcati…

**Figure 7.** Figure 7: The number of particles N (a) and IPR χ (b) are plotted against the chemical potential µ, for families of nonlinear modes that emerge in gap B in an asymmetric quasiperiodic potential described by Eq. (13) with a nonzero phase shift θ ≈ 1.7π. The yellow circles in (a…

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

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W. Su, Z. Lin, C. Li, C. Huang, Nonlinear localized gap modes in width-modulated Fibonacci lattices, Results in Physics, 40, 105877 (2022)

work page 2022

[2] [1]

Most of these facts have been discussed in previous studies on bichromatic quasiperi- odic potentials (see [ 48–50, 52–62])

Here, we summarize several facts about the linear spectrum that are relevant to our study. Most of these facts have been discussed in previous studies on bichromatic quasiperi- odic potentials (see [ 48–50, 52–62]). (i) The spectrum of the linear eigenvalue problem ( 4) is bounded from below and discrete, due to the 4 periodic boundary conditions. We enum...

[3] [2]

In this respect the bichromatic lattice given by Eq

in fact corresponds to a special situation, because (i) it is symmetric with respect to the origin x = 0, and (ii) it achieves its maximum (both local and global) at x = 0. In this respect the bichromatic lattice given by Eq. (

[4] [3]

In a more general situation, the bichromatic lattice comprises a combination of mutually shifted incommensurate dependences

resembles a double-well potential with the bar- rier situated at x = 0. In a more general situation, the bichromatic lattice comprises a combination of mutually shifted incommensurate dependences. This more general case will be discussed below in Sec. IV E. To put our work into proper context, we notice that the potential ( 2) diﬀers from the bichromatic ...

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J. B. Reeves, B. Gadway, T. Bergeman, I. Danshita, and D. Schneble, Superﬂuid Bloch dynamics in an incommen- 11 surate optical lattice, New J. Phys. 16, 065011 (2014)

work page 2014

[6] [5]

Its period is equal to Lq := πq

is periodic. Its period is equal to Lq := πq . The diﬀerence between Vq(x) and V (x) remains small only in a bounded spatial domain. As a result, it becomes relevant to restrict our consideration to a ﬁnite domain by imposing the periodic boundary conditions. Speciﬁcally, we require Ψ( −πq/ 2,t ) = Ψ( πq/ 2,t ) [ ψ (−πq/ 2) = ψ (πq/ 2)]. (7) The choice of...

[7] [6]

24 ≲ µ ≲ − 0

In Table I this sit- 5 p/q = 55/ 34 p/q = 89/ 55 p/q = 144/ 89 p/q = 233/ 144 gap A: − 0. 24 ≲ µ ≲ − 0. 23 s-a/s pair s-a/s pair boundary new s-a/s pair gap B: − 0. 22 ≲ µ ≲ − 0. 17 boundary s-a/s pair s-a/s pair boundary gap C: − 0. 15 ≲ µ ≲ − 0. 02 symmetric mode for any p/q gap D: − 0. 15 ≲ µ ≲ − 0. 02 s-a/s pair s-a/s pair boundary new s-a/s pair Tabl...

[8] [7]

ac- tually represent Bloch functions of the band structure associated with the periodic potential Vq(x). This fact may be useful for for interpreting the results; (iii) Peri- odic boundary conditions are ideal for eﬃcient numerical methods used to solve linear eigenproblems of the form ( 4). To solve the linear eigenproblem ( 4) with the pe- riodic potent...